Forced Vibrations of Thin, Elastic, Rectangular Plates with Edges Elastically Restrained Against Rotation

1977 ◽  
Vol 21 (01) ◽  
pp. 24-29
Author(s):  
E. A. Susemihl ◽  
P. A. A. Laura
Keyword(s):  
Bending Moment ◽  
Forced Vibrations ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Edge Conditions ◽  
Approach Infinity ◽  
Square Plate ◽  
Elastic Rectangular Plate ◽  
Elastically Restrained ◽  
The Galerkin Method

Polynomial coordinate functions and the Galerkin method are used to determine the response of a thin, elastic, rectangular plate with edges elastically restrained against rotation and subjected to sinusoidal excitation. It is shown that when the flexibility coefficients approach infinity (simply supported edge conditions) the calculated results practically coincide with the exact solution in the case of a square plate when four terms of the expansion are used. Dynamic displacement and bending moment amplitudes are tabulated for different length-to-width ratios, flexibility coefficients, and frequency values.

Bending of Elastically Restrained Rectangular Plates Under Uniform Pressure

1958 ◽  
Vol 62 (575) ◽  
pp. 834-836 ◽  
Author(s):  
C. Lakshmi Kantham
Keyword(s):  
Natural Frequencies ◽  
Unit Length ◽  
Rectangular Plates ◽  
Uniform Pressure ◽  
Simply Supported ◽  
Edge Conditions ◽  
Definition Of ◽  
Elastically Restrained ◽  
Clamped Edges ◽  
Clamped Edge

In the bending and vibration of plates it is found that the values of maximum deflection and natural frequencies, respectively, vary considerably from the simply-supported to clamped edge conditions. For an estimation of these characteristics in the intermediate range a generalised boundary condition may be assumed, of which the simply-supported and clamped edges become limiting cases. While Bassali considers the ratio of edge moment to the cross-wise moment as a constant, Newmark, Lurie and Klein and other investigators, in their analyses of various structures, consider that moment and slope at an end are proportional.Here the definition of elastic restraint as given by Timoshenko, α=βM, is followed, where α is the slope at any edge, M the corresponding edge moment per unit length while β is the elastic restraint factor. β→0 and β→∞ represent the two limiting cases of simply-supported and clamped edge conditions.

Some Observations on the Compressive Buckling of Rectangular Plates

1963 ◽  
Vol 14 (1) ◽  
pp. 17-30 ◽  
Author(s):  
W. H. Wittrick
Keyword(s):  
Rate Of Change ◽  
Infinite Strip ◽  
Rectangular Plates ◽  
Buckling Mode ◽  
Simply Supported ◽  
Side Ratio ◽  
Buckling Stress ◽  
Stress Coefficient ◽  
Wide Range ◽  
Elastically Restrained

SummaryThe problem considered is the buckling of a rectangular plate under uniaxial compression. The ends may be either both clamped, both simply-supported or a mixture of the two. The sides may be elastically restrained against both deflection and rotation with any stiffnesses whatsoever. It is shown that the curve of buckling stress coefficient versus side ratio can be deduced in a simple manner from that of a plate with the same end conditions but with both sides simply-supported, provided only that the buckling stress coefficient and wavelength for an infinite strip with the same side conditions are known. Some correlations between the curves for the three types of end condition are discussed. It is also shown that if, for some given side ratio, the buckling mode is known, then it is always possible to deduce the rate of change of buckling stress coefficient with side ratio at that point. The argument is based upon an assumption which is shown to give very accurate results in a wide range of cases.

The Plastic Buckling of Axially Compressed Square Tubes

1992 ◽  
Vol 59 (2) ◽  
pp. 276-282 ◽  
Author(s):  
S. Li ◽  
S. R. Reid
Keyword(s):  
Deformation Theory ◽  
Buckling Analysis ◽  
Rectangular Plates ◽  
Plastic Buckling ◽  
Buckling Mode ◽  
Important Conclusion ◽  
Simply Supported ◽  
Edge Conditions ◽  
Crushing Behavior ◽  
Incremental Theory Of Plasticity

A plastic buckling analysis for axially compressed square tubes is described in this paper. Deformation theory is used together with the realistic edge conditions for the panels of the tube introduced in our previous paper (Li and Reid, 1990), referred to hereafter as LR. The results obtained further our understanding of a number of problems related to the plastic buckling of axially compressed square tubes and simply supported rectangular plates, which have remained unsolved hitherto and seem rather puzzling. One of these is the discrepancy between experimental results and the results of plastic buckling analysis performed using the incremental theory of plasticity and the unexpected agreement between the results of calculations based on deformation theory for plates and experimental data obtained from tests conducted on tubes. The non-negligible difference between plates and tubes obtained in the present paper suggests that new experiments should be carried out to provide a more accurate assessment of the predictions of the two theories. Discussion of the results herein also advances our understanding of the compact crushing behavior of square tubes beyond that given in LR. An important conclusion reached is that strain hardening cannot be neglected for the plastic buckling analysis of square tubes even if the degree of hardening is small since doing so leads to an unrealistic buckling mode.

VIBRATION AND BUCKLING OF SS-F-SS-F RECTANGULAR PLATES LOADED BY IN-PLANE MOMENTS

2001 ◽  
Vol 01 (04) ◽  
pp. 527-543 ◽  
Author(s):  
JAE-HOON KANG ◽  
ARTHUR W. LEISSA
Keyword(s):  
Beam Theory ◽  
Free Edge ◽  
Free Vibrations ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
One Dimensional ◽  
Simply Supported ◽  
Free Vibration Frequency ◽  
Edge Conditions

This paper presents exact solutions for the free vibrations and buckling of rectangular plates having two opposite, simply supported edges subjected to linearly varying normal stresses causing pure in-plane moments, the other two edges being free. Assuming displacement functions which are sinusoidal in the direction of loading (x), the simply supported edge conditions are satisfied exactly. With this the differential equation of motion for the plate is reduced to an ordinary one having variable coefficients (in y). This equation is solved exactly by assuming power series in y and obtaining its proper coefficients (the method of Frobenius). Applying the free edge boundary conditions at y=0, b yields a fourth order characteristic determinant for the critical buckling moments and vibration frequencies. Convergence of the series is studied carefully. Numerical results are obtained for the critical buckling moments and some of their associated mode shapes. Comparisons are made with known results from less accurate one-dimensional beam theory. Free vibration frequency and mode shape results are also presented. Because the buckling and frequency parameters depend upon the Poisson's ratio (ν), results are shown for 0≤ν≤0.5, valid for isotropic materials.

Analytical Buckling Loads of Composite Rectangular Plates with Vertical and Rotational Springs Along the Edges

2020 ◽  
pp. 2043002
Author(s):  
Joseph Tenenbaum ◽  
Moshe Eisenberger
Keyword(s):  
Boundary Conditions ◽  
Bending Moment ◽  
Composite Plates ◽  
Static Solution ◽  
Published Data ◽  
Rectangular Plates ◽  
Buckling Loads ◽  
Edge Conditions ◽  
In Series ◽  
Edge Force

In this research, a new analytical solution is used for finding the buckling loads of rectangular plates with vertically and rotationally restrained edges. The solution method in this study is based on the development of a static solution for a plate. The solution is obtained in series form, and the coefficients are solved to match the edge conditions. The solution fits all the combinations of possible boundary conditions, of the deflection, slope, shear force and bending moment along the edges of the plate. In the case of springs, the edge force and moment boundary conditions are modified to include these effects. Any number of edges, from one to four, with both types of stiffening springs can be solved. Using this new method, the exact buckling loads and modes are found. The results are verified with published data, and many new cases are presented for uni-axially and bi-axially loaded isotropic, orthotropic, and composite plates.

Exact Solutions for the Free Vibrations and Buckling of Rectangular Plates With Linearly Varying In-Plane Loading

2001 ◽  
Author(s):  
Arthur W. Leissa ◽  
Jae-Hoon Kang
Keyword(s):  
Solution Procedure ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Buckling Loads ◽  
Simply Supported ◽  
Edge Conditions ◽  
Elastically Supported

Abstract An exact solution procedure is formulated for the free vibration and buckling analysis of rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying normal stresses. The other two edges may be clamped, simply supported or free, or they may be elastically supported. The transverse displacement (w) is assumed as sinusoidal in the direction of loading (x), and a power series is assumed in the lateral (y) direction (i.e., the method of Frobenius). Applying the boundary conditions yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to obtain adequate convergence for accurate vibration frequencies and buckling loads, as is demonstrated by two convergence tables. Some interesting and useful results for vibration frequencies and buckling loads, and their mode shapes, are presented for a variety of edge conditions and in-plane loadings, especially pure in-plane moments.

scholarly journals Frequency Equations and Modes of Free Vibrations of Rectangular Plates with Various Edge Conditions

1994 ◽  
Vol 208 (5) ◽  
pp. 307-319 ◽  
Author(s):  
C W Bert ◽  
M Malik
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Free Vibrations ◽  
Edge Condition ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Edge Conditions ◽  
Classical Boundary ◽  
Free Edges

This paper considers linear free vibrations of thin isotropic rectangular plates with combinations of the classical boundary conditions of simply supported, clamped and free edges and the mathematically possible condition of guided edges. The total number of plate configurations with the classical boundary conditions are known to be twenty-one. The inclusion of the guided edge condition gives rise to an additional thirty-four plate configurations. Of these additional cases, twenty-one cases have exact solutions for which frequency equations in explicit or transcendental form may be obtained. The frequency equations of these cases are given and, for each case, results of the first nine mode frequencies are tabulated for a range of the plate aspect ratios.

scholarly journals Series-Based Solution for Analysis of Simply Supported Rectangular Thin Plate with Internal Rigid Supports

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Abubakr E. S. Musa ◽  
Husain J. Al-Gahtani
Keyword(s):  
Analytical Solution ◽  
Bending Moment ◽  
Accurate Solution ◽  
Rectangular Plates ◽  
Compatibility Conditions ◽  
Element Analysis ◽  
Simply Supported ◽  
The Finite Element Analysis ◽  
Rigid Supports ◽  
The Given

In this study, Navier’s solution for the analysis of simply supported rectangular plates is extended to consider rigid internal supports. The proposed method offers a more accurate solution for the bending moment at the critical section and therefore serves as a better analytical solution for design purposes. To model the plate-support interaction, the patched areas representing the contact between the plate and supports are divided into groups of cells. The unknown internal reactions at the centers of the divided cells are obtained by satisfying the compatibility conditions at the centers of the cells. Three numerical examples are presented to demonstrate the accuracy of the proposed analytical solution. The given examples reveal good agreements with those obtained by the finite element analysis. In addition, they show the advantage of the new solution as compared to the existing analytical solution which inaccurately estimates the location and magnitude of the maximum bending moment.

Flexural problems of circular ring plates and sectorial plates. I

1960 ◽  
Vol 56 (1) ◽  
pp. 75-95 ◽  
Author(s):  
W. A. Bassali ◽  
M. A. Gorgui
Keyword(s):  
Circular Ring ◽  
Concentric Circle ◽  
Thin Plates ◽  
Constant Thickness ◽  
Concentrated Load ◽  
Simply Supported ◽  
General Line ◽  
Special Cases ◽  
Edge Conditions ◽  
Elastically Restrained

ABSTRACTIn this paper explicit expressions in closed forms are first obtained for the complex potentials and deflexion at any point of a circular annular plate under various edge conditions when the plate is acted upon by general line loadings distributed along the circumference of a concentric circle. These solutions are then used to discuss the bending of a circular plate with a central hole under a concentrated load or a concentrated couple acting at any point of the plate. Solutions for singularly loaded sectorial plates bounded by two arcs of concentric circles and two radii are also derived when the plate is simply supported along the straight edges. The boundary conditions along the circular edges include the cases of a free boundary as well as the elastically restrained boundary which covers the usual rigidly clamped and simply supported boundaries as special cases. The usual restrictions relating to the small deflexion theory of thin plates of constant thickness are assumed. Limiting forms of the resulting solutions are investigated.

Stability of Rectangular Plates Under Shear and Bending Forces

1936 ◽  
Vol 3 (4) ◽  
pp. A131-A135 ◽  
Author(s):  
Stewart Way
Keyword(s):  
Critical Load ◽  
Rectangular Plate ◽  
Bending Moment ◽  
The Other ◽  
Rectangular Plates ◽  
Bending Stress ◽  
Simply Supported ◽  
Tension And Compression ◽  
The Way

Abstract The author first discusses the problem of a plane, simply supported rectangular plate loaded by shearing forces in the plane of the plate on all four edges. There are two stiffeners attached one third and two thirds of the way along the plate. The critical load is calculated for various stiffener rigidities. Also, the rigidity necessary to keep the stiffeners straight when the plate buckles is found. This stiffener rigidity is found to be slightly larger than that necessary for a plate with one stiffener and the same panel dimensions as the plate with two stiffeners. The second problem discussed by the author is that of a plane, simply supported rectangular plate loaded by uniformly distributed edge shearing forces in the plane of the plate and linearly distributed tension and compression in the plane of the plate at the ends. The end forces vary from tension hσo, at one corner to—hσo, at the other corner, so that their resultant is a bending moment. The presence of the edge shearing forces is found to diminish the critical bending stress in this case. Calculations are made for various magnitudes of bending and shearing forces for plates of various proportions.

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